Optimal. Leaf size=52 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]
[Out]
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Rubi [A] time = 0.0570127, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 5.9992, size = 46, normalized size = 0.88 \[ - \frac{4 b \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15 c^{2} x^{\frac{3}{2}}} + \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 c \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.024281, size = 41, normalized size = 0.79 \[ \frac{2 \sqrt{x (b+c x)} \left (-2 b^2+b c x+3 c^2 x^2\right )}{15 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 33, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,cx+2\,b \right ) }{15\,{c}^{2}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.723939, size = 41, normalized size = 0.79 \[ \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b}}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236252, size = 70, normalized size = 1.35 \[ \frac{2 \,{\left (3 \, c^{3} x^{4} + 4 \, b c^{2} x^{3} - b^{2} c x^{2} - 2 \, b^{3} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{x \left (b + c x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210571, size = 46, normalized size = 0.88 \[ \frac{4 \, b^{\frac{5}{2}}}{15 \, c^{2}} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b\right )}}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="giac")
[Out]